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Blow-up of solutions to some classes of ill-posed operator equations and a growth quantification result

  • Jehan R. Abbas ORCID logo , Mercy Grace C. Flores ORCID logo , Noli N. Reyes and Louie John D. Vallejo ORCID logo EMAIL logo
Published/Copyright: February 10, 2025
Journal of Inverse and Ill-posed Problems
From the journal Journal of Inverse and Ill-posed Problems Volume 33 Issue 3

Abstract

Let ℋ and 𝒦 be real Hilbert spaces, let T : ℋ → 𝒦 be a continuous operator with a non-closed range, and let y ∈ ran ⁡ T ¯ ∖ ran ⁡ T . In the case where ℋ is finite-dimensional, or T is linear, or ℋ = 𝒦 and Id - T is nonexpansive, we show that the approximate solutions to T ⁢ x = y blow up. We also show that the blow-up of approximate solutions is not necessary by constructing a continuous mapping T : L 2 ⁢ ( [ 0 , 1 ] ) → L 2 ⁢ ( [ 0 , 1 ] ) , a vector y ∈ ran ⁡ T ¯ ∖ ran ⁡ T , and a bounded sequence of approximate solutions to T ⁢ x = y . Finally, if T : ℋ → 𝒦 is compact and self-adjoint, and y ∈ ran ⁡ T ¯ ∖ ran ⁡ T , we obtain a growth quantification result via an inequality of the form φ y ⁢ ( ∥ T ⁢ x - y ∥ ) ⋅ ∥ x ∥ ≥ C where φ y is a nondecreasing function and C is a positive constant.

Keywords: Ill-posed operator equations; blow-up of approximate solutions; compact self-adjoint operators; quantification of growth
MSC 2020: 40E10; 47A52; 47A58; 47A63; 47B07; 47J06

Funding statement: Jehan R. Abbas’s and Mercy Grace C. Flores’s work were supported by the Department of Science and Technology – Science Education Institute under the Accelerated Science and Technology Human Resource Development Program (DOST-ASTHRDP). Louie John D. Vallejo’s work was supported in part the OVCRD PhD Incentive Award and in part by a Research Grant under the Computational Research Laboratory of the Institute of Mathematics, UP Diliman.

Acknowledgements

Noli N. Reyes passed away before the submission of this article. The authors are grateful to the anonymous referees who gave constructive and helpful comments and suggestions to improve this manuscript.

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Received: 2024-03-17
Revised: 2024-10-30
Accepted: 2025-01-29
Published Online: 2025-02-10
Published in Print: 2025-06-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. An inverse problem for non-selfadjoint Dirac operator with transmission conditions at finite interior points
  3. Inverse Sturm–Liouville problem with polynomials in the boundary condition and multiple eigenvalues
  4. Blow-up of solutions to some classes of ill-posed operator equations and a growth quantification result
  5. On an inverse problem with high-order overdetermination conditions
  6. Two regularization methods for identifying the source term of Caputo–Hadamard type time fractional diffusion-wave equation
  7. Integrating the probe and singular sources methods: III. Mixed obstacle case
  8. Stability estimate and the Tikhonov regularization method for the Kuramoto–Sivashinsky equation backward in time
  9. Continuation of solutions of elliptic systems from discrete sets with application to geometry of surfaces
https://doi.org/10.1515/jiip-2024-0020
Keywords for this article
Ill-posed operator equations; blow-up of approximate solutions; compact self-adjoint operators; quantification of growth

Articles in the same Issue

  1. Frontmatter
  2. An inverse problem for non-selfadjoint Dirac operator with transmission conditions at finite interior points
  3. Inverse Sturm–Liouville problem with polynomials in the boundary condition and multiple eigenvalues
  4. Blow-up of solutions to some classes of ill-posed operator equations and a growth quantification result
  5. On an inverse problem with high-order overdetermination conditions
  6. Two regularization methods for identifying the source term of Caputo–Hadamard type time fractional diffusion-wave equation
  7. Integrating the probe and singular sources methods: III. Mixed obstacle case
  8. Stability estimate and the Tikhonov regularization method for the Kuramoto–Sivashinsky equation backward in time
  9. Continuation of solutions of elliptic systems from discrete sets with application to geometry of surfaces
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